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Properties

Label 26T27
Degree $26$
Order $3042$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:(C_3\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(26, 27);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{13}^2:(C_3\times S_3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,16,3,21,9,23)(2,25,6,22,5,26)(4,17,12,24,10,19)(7,18,8,14,11,15)(13,20), (1,6,8)(2,9,4)(3,12,13)(7,11,10)(14,16,18,20,22,24,26,15,17,19,21,23,25)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T39, 39T40

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 13, 13 $ $18$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $18$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $18$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $18$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $(14,17,20,23,26,16,19,22,25,15,18,21,24)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
$ 13, 13 $ $9$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $18$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $9$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $9$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $18$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$
$ 13, 13 $ $9$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,21)(15,22,17)(16,25,26) (20,24,23)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,15)(16,17,20)(18,23,25) (19,26,21)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,16)(15,24,25)(18,20,26) (19,23,22)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,18)(16,21,23)(17,24,19) (22,26,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 2,11)( 3, 7, 4)( 5,12,10)( 6, 8,13)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 3, 8)( 2,12,11)( 5,13, 7)( 6, 9,10)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 5, 2)( 3,10, 8)( 4, 6,11)( 9,12,13)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 3, 3, 3, 3, 1 $ $78$ $39$ $( 1, 9, 3)( 2, 5, 6)( 4,10,12)( 7,11, 8)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 6, 6, 6, 6, 2 $ $507$ $6$ $( 1,16, 3,21, 9,23)( 2,25, 6,22, 5,26)( 4,17,12,24,10,19)( 7,18, 8,14,11,15) (13,20)$
$ 26 $ $117$ $26$ $( 1,20,11,17, 8,14, 5,24, 2,21,12,18, 9,15, 6,25, 3,22,13,19,10,16, 7,26, 4,23 )$
$ 26 $ $117$ $26$ $( 1,25, 4,15, 7,18,10,21,13,24, 3,14, 6,17, 9,20,12,23, 2,26, 5,16, 8,19,11,22 )$
$ 26 $ $117$ $26$ $( 1,14, 9,22, 4,17,12,25, 7,20, 2,15,10,23, 5,18,13,26, 8,21, 3,16,11,24, 6,19 )$
$ 26 $ $117$ $26$ $( 1,21, 7,14,13,20, 6,26,12,19, 5,25,11,18, 4,24,10,17, 3,23, 9,16, 2,22, 8,15 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $39$ $2$ $( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,26) (12,14)(13,15)$
$ 6, 6, 6, 6, 2 $ $507$ $6$ $( 1,19, 2,22,11,23)( 3,25, 7,24, 4,15)( 5,18,12,26,10,20)( 6,21, 8,14,13,16) ( 9,17)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $3042=2 \cdot 3^{2} \cdot 13^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  3042.e
magma: IdentifyGroup(G);
 
Character table:    35 x 35 character table

magma: CharacterTable(G);